Yeah, they'll figure it out. Most animals have little more complex hearing circuits than we do,and they've evolved to pick more and different things out of sounds than we do. When dog hears his name, it's usually said with the same type of inflection as the human addresses it specifically to the dog. That inflection is almost as important as the actual word, as subconsciously we'r probably making that inflection slightly different for each animal we address, no matter what we call them. Stay consistent with the new name. Address the dog the same way with the new name as with the old name, and it should be a quick process. Chances are she'll probably retain the old name pretty well, also.
We have five parrots, and each of them has a proper name that hardly ever gets used because we use love of a couple different nicknames for each of them. But there's still very little confusion over who is being addressed no matter what nickname is being used. They all seem to know their slave names pretty well.
What I'd REALLY like to know is what their names are for each other (because it's been confirmed that they have specific names within a flock) and if those names have any relationship to their earth names. Also, since we have a mixed flock (cockatoos and eclectus) I wonder what type of interspecies communication they've developed after years together, or if there already was some similar communication patterns since both species exist on some of the same Indonesian islands.
And anyone that doesn't think inflection is a big deal, spend a little time with a parrot. Even one with a limited vocabulary. While it's pretty easy for them to pick up a basic vocabulary contextually, it's crazy to watch them take the limited words they know and inflect the hell out of them to get a message across. Our bonded cockatoo couple wants to go to bed first every night. So around the same time the male will see us and give us a quick, informational "Good night, kids." which means it's time for his wife to come in from whatever part of the habitat she's in, for them to get their pre-bed treats, hop in their cage and get covered up.
If they, for whatever reason, want to go to bed early—maybe it's overcast and got darker earlier—the "Good night kids?" is clearly delivered as a question rather than as a statement.
And if for some reason we should go into the bird room late at night once everyone is in bed—maybe one of the other birds as up with us watching tv—you will hear absolutely the most passive aggressive "Good NIGHT, kids." delivered in an angry whisper from their covered cage.
So, yeah. Long story short: Animals are awesome with sounds and understand how we use them to communicate almost as well as we do. Don't sweat it.